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Chapter 2: Problem 31

Find the transpose of each matrix. \(\left[\begin{array}{llll}3 & 2 & -1 & 5\end{array}\right]\)

Short Answer

Expert verified
The transpose of the given matrix A = \(\left[\begin{array}{llll}3 & 2 & -1 & 5\end{array}\right]\) is A^T = \(\begin{bmatrix} 3 \\ 2 \\ -1 \\ 5 \end{bmatrix}\).

Step by step solution

01

1. Identify the Given Matrix

The given matrix has only one row and four columns, so we can represent it as A = \(\left[\begin{array}{llll}3 & 2 & -1 & 5\end{array}\right]\).
02

2. Determine the Size of the Transpose Matrix

The transpose of a matrix is obtained by swapping the rows and columns of the given matrix. In this case, the given matrix is a (1x4) matrix. The transpose of this matrix will have size (4x1) - it will have 4 rows and 1 column.
03

3. Find the Transpose

For a given matrix element A(i, j), the corresponding element in the transpose matrix A^T(i, j) is found using the following rule: A^T(i, j) = A(j, i). So in our case, we have: A^T(1, 1) = A(1, 1) = 3 A^T(2, 1) = A(1, 2) = 2 A^T(3, 1) = A(1, 3) = -1 A^T(4, 1) = A(1, 4) = 5 So, the transpose of A is given by: A^T = \(\begin{bmatrix} 3 \\ 2 \\ -1 \\ 5 \end{bmatrix}\)

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Most popular questions from this chapter

Bob, a nutritionist who works for the University Medical Center, has been asked to prepare special diets for two patients, Susan and Tom. Bob has decided that Susan's meals should contain at least \(400 \mathrm{mg}\) of calcium, \(20 \mathrm{mg}\) of iron, and \(50 \mathrm{mg}\) of vitamin \(\mathrm{C}\). whereas Tom's meals should contain at least \(350 \mathrm{mg}\) of calcium, \(15 \mathrm{mg}\) of iron, and \(40 \mathrm{mg}\) of vitamin \(\mathrm{C}\). Bob has also decided that the meals are to be prepared from three basic foods: food \(\mathrm{A}\), food \(\mathrm{B}\), and food \(\mathrm{C}\). The special nutritional contents of these foods are summarized in the accompanying table. Find how many ounces of each type of food should be used in a meal so that the minimum requirements of calcium, iron, and vitamin \(\mathrm{C}\) are met for each patient's meals. $$ \begin{array}{lccc} \hline && {\text { Contents (mg/oz) }} & \\ & \text { Calcium } & \text { Iron } & \text { Vitamin C } \\ \hline \text { Food A } & 30 & 1 & 2 \\ \hline \text { Food B } & 25 & 1 & 5 \\ \hline \text { Food C } & 20 & 2 & 4 \\ \hline \end{array} $$

Write the given system of linear equations in matrix form. $$ \begin{array}{l} 2 x= 7 \\ 3 x-2 y=12 \end{array} $$

The problems in exercise correspond to those in exercises 15-27, Section 2.1. Use the results of your previous work to help you solve these problems. A dietitian wishes to plan a meal around three foods. The percent of the daily requirements of proteins, carbohydrates, and iron contained in each ounce of the three foods is summarized in the following table: $$\begin{array}{lccc} \hline & \text { Food I } & \text { Food II } & \text { Food III } \\ \hline \text { Proteins }(\%) & 10 & 6 & 8 \\ \hline \text { Carbohydrates }(\%) & 10 & 12 & 6 \\ \hline \text { Iron }(\%) & 5 & 4 & 12 \\ \hline \end{array}$$ Determine how many ounces of each food the dietitian should include in the meal to meet exactly the daily requirement of proteins, carbohydrates, and iron \((100 \%\) of each).

Find the value(s) of \(k\) such that $$ A=\left[\begin{array}{rrr} 1 & 0 & 1 \\ -2 & 1 & k \\ -1 & 2 & k^{2} \end{array}\right] $$ has an inverse. Find the value(s) of \(k\) such that the augmented matrix \([A \mid I]\) can be reduced to the form \([I \mid B]\).

Solve the system of linear equations using the Gauss-Jordan elimination method. $$ \begin{array}{r} 2 x_{1}-x_{2}-x_{3}=0 \\ 3 x_{1}+2 x_{2}+x_{3}=7 \\ x_{1}+2 x_{2}+2 x_{3}=5 \end{array} $$
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